Theorem 12.10.43.A528C756F776 Let $0\leq f\in L^1(\mathrm{d}V)$ be nonzero with $b=b_\omega (f)>0$, and put $p=f^{1/n}$ and $R=\rho _\omega (p)=b^{-1/n}$. Suppose that there are smooth strictly positive $T_ j\in \mathscr G_\omega $, with associated probability measures $\nu _ j$, such that
\[ d_ j=D_ p(T_ j)\longrightarrow R,\qquad \mathrm{d}\nu _ j=d_ j^{-1}p k_{T_ j}\, \mathrm{d}V, \]
and a constant $C$ satisfying
797F5974140E
12.10.44
\begin{equation} \int _ X(-v)\, \mathrm{d}\nu _ j\leq C \quad \text{for all }j\text{ and all }v\in \mathrm{PSH}(X,\omega ) \text{ with }\sup _ Xv=0. \end{equation}
Then, for every $\lambda >0$,
\[ b\leq c_\lambda \leq b\mathrm{e}^{C\lambda },\qquad \Phi _ f-C\leq v_\lambda \leq 0. \]
Suppose in addition that $P_{\delta \omega }(\Phi _ f)$ is not identically $-\infty $ for some $0<\delta <1$. Whenever $\lambda _ k\downarrow 0$ and $v_{\lambda _ k}\to v$ in $L^1(X)$, the convergence holds in capacity and
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v)=bf\, \mathrm{d}V, \qquad \Phi _ f-C\leq v\leq \Phi _ f. \]
In particular an exact threshold solution exists.
Proof.
Since $\mathrm{MA}^{\mathrm{ext}}_\omega (v_\lambda )=c_\lambda \mathrm{e}^{\lambda v_\lambda }f\, \mathrm{d}V$, the trace inequality of lemma 13.14.12 and Stokes' theorem give, for each $j$ and $\lambda >0$,
\[ 1\geq c_\lambda ^{1/n}d_ j \int _ X\mathrm{e}^{\lambda v_\lambda /n}\, \mathrm{d}\nu _ j \geq c_\lambda ^{1/n}d_ j\mathrm{e}^{-C\lambda /n}. \]
The second inequality follows from Jensen's inequality and (12.10.44). Letting $j\to \infty $ and using $R=b^{-1/n}$, we obtain the upper bound for $c_\lambda $. The lower bound and the comparison with every normalized threshold subsolution were proved in corollary 15.35.3; applying that comparison to $\Phi _ f$, we obtain
\[ v_\lambda \geq \Phi _ f-\lambda ^{-1}\log (c_\lambda /b) \geq \Phi _ f-C. \]
Now let $v_{\lambda _ k}\to v$ in $L^1$. By proposition 15.28.6, the limit $v$ lies in $\mathcal S_ f$, hence below $\Phi _ f$. The lower bound passes to the limit, so $\Phi _ f-C\leq v\leq \Phi _ f$; in particular $v\sim \Phi _ f$ in the sense of definition 16.7.1. For all $k$ with $\lambda _ k\leq 1$,
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ k}) =c_{\lambda _ k}\mathrm{e}^{\lambda _ k v_{\lambda _ k}}f\, \mathrm{d}V \leq b\mathrm{e}^{C}f\, \mathrm{d}V. \]
The functions $|\mathrm{e}^{v_{\lambda _ k}}-\mathrm{e}^ v|$ are uniformly bounded and tend to zero in $\mathrm{d}V$-measure, so their integrals against the fixed $L^1$ density $f$ tend to zero. Hence the error term in (15.9.2) tends to zero, as required by proposition 15.9.3 with $\varphi =\Phi _ f$. The additional hypothesis $v\preceq \Phi _ f$ of that proposition holds since $v\leq \Phi _ f$, and the condition on $P_{\delta \omega }(\Phi _ f)$ is assumed. That proposition therefore gives convergence in capacity.
The continuity of damped measures under convergence in capacity, theorem 13.15.9, now identifies the equation without a mass-conservation argument. Write $g_ k=c_{\lambda _ k}\mathrm{e}^{\lambda _ k v_{\lambda _ k}}f$. We know $g_ k\to bf$ in $L^1$ and $0\leq g_ k\leq b\mathrm{e}^ Cf$ for large $k$. Hence
\[ \mathrm{e}^{v_{\lambda _ k}}\mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ k}) =\mathrm{e}^{v_{\lambda _ k}}g_ k\, \mathrm{d}V \longrightarrow \mathrm{e}^ v bf\, \mathrm{d}V \]
in total variation, while theorem 13.15.9 gives the weak limit $\mathrm{e}^ v\mathrm{MA}^{\mathrm{ext}}_\omega (v)$. Dividing by $\mathrm{e}^ v$ off the pole set of $v$, which neither measure charges, gives the equation and the finiteness of the mass. Finally, compactness of normalized $\omega $-plurisubharmonic functions provides an $L^1$ cluster limit along any sequence $\lambda _ k\downarrow 0$, which proves existence.
$\square$
Pome
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